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I need to find the temperature distribution of a square plate using the Laplace equation by using FDM:

$$ \frac{d^2T}{dx^2} + \frac{d^2T}{dy^2} = 0$$

But there is a heat flux entering from the top boundary $q = 10^5 W/m^2$. How do I incorporate this heat flux into the problem? Should I:

  1. Add it as a boundary condition by considering area $A=1$ and make the top boundary $10^5$? I saw something of this sort in an FVM textbook but dimensionally, it doesn't make sense to me.

  2. Consider the 1-D heat flux equation:

$$ q = -k \frac{dT}{dy} $$

and discretize it as $$\frac{T_{i} - T_{i-1}}{dy} = -q/k$$

and incorporate this into every x-iteration?

This method makes sense to me but when I tried it, the contour plot showed the temperature in negatives as so: enter image description here

If I am adding a heat flux $10^5$, why is the temperature in negatives?

  1. Another method?
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    $\begingroup$ At top boundary, $q = -k \nabla T \cdot n > 0$, then heat is flowing out of the top boundary. You have $T=0$ on other three sides, so thinking physically, negative temperature is not wrong in this case. $\endgroup$
    – cfdlab
    Mar 8, 2021 at 7:24
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    $\begingroup$ You can use ghost points, in this case. That is, you introduce (conceptually) an additional row of points $\Delta y$ above the first one, and you compute the values of the temperature along this line such that a first-order discretisation of the heat flux (using these ghost points and the points of the first row) yields the prescribed flux $\phi$. For instance ($i$ is the index along $x$, $j$ for $y$): $\pm\lambda\frac{T_{ghost~up,j} - T_{i,j}}{\Delta y} = \phi_j$, from which you can compute $T_{ghost~up,j}$ and use it to compute the discrete laplacin in the first (real) row of points. $\endgroup$
    – Laurent90
    Mar 8, 2021 at 8:33
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    $\begingroup$ Another possibility that may be easier to implement is to put a source term in the guard cells, and enforce zero gradient of the temperature on the external sides of the guard cells. $\endgroup$
    – Maxim Umansky
    Mar 8, 2021 at 9:17
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    $\begingroup$ Your question is confusing the modeling part and the numerics. First derive a model that consists of the PDE and its boundary conditions. Then you can wonder how to discretize it. $\endgroup$
    – Wolfgang Bangerth
    Mar 8, 2021 at 9:47
  • $\begingroup$ Thanks everyone for your inputs. I was able to solve it. @WolfgangBangerth yes the PDE is the Laplace equation and the boundary conditions are zero on all sides. $\endgroup$
    – justauser
    Mar 8, 2021 at 11:58

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Because a heat flux has a direction, and from what you are describing, you are adding a heat flux in the normal direction -- out of the domain. So if you started with a constant temperature plane and they draw heat out of it, you'd end up with a negative temperature.

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